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Differential Calculus

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13 Mar 2010 13:23:12 IST

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Increasing Decreasing Function

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(x) = 2log(x-2) - x^2 + 4x + 1f(x) is defined for all x>2f '(x) = -2(x-1)(x-3) / (x-2)For stationary points, f '(x) = 0Then what will the values of x And also what will be the intervals in which f(x) is increasing or decreasingPlease explain .

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Deepak Aggarwal

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13 Mar 2010 20:36:20 IST

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f'(x) = [ 2 / (x-2) ] - 2x + 4

f'(x) = 0

So, 2 / (x - 2) = 2x - 4

or, (x - 2)² = 1

or, x² - 4x + 3 = 0

or, (x - 1)(x - 3) = 0

or, x = 3 or x = 4

Also, f'(x) = (2) ( 1 - (x - 2)² ) / (x - 2) = (-2) (x - 3) (x - 4) / (x - 2)

So, function decreases in (-infinity,-3)U(4,infinity) and increases in (3,4).

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